There are two basic ways you should use Google Analytics: performing analysis on specific improvement initiatives, and monitoring the overall 'health' of your website. This article focuses on website 'health'; what metrics you should monitor on a regular basis for problems.
Focus on these core metrics first.
Visits and Visitors are the most basic metric you can measure. You'll probably want to do a historical comparison, e.g. month-over-month or quarter-over-quarter, to see if your website is growing. For a definition of Visits and Visitors, see Tracking Terminology.
The traffic sources report tells you where your traffic has come from. If you do a month-over-month (or quarter-over-quarter) comparison, you can easily ferret out if you have any health issues with your traffic sources. For example, if you saw your google / cpc traffic drop considerably month-to-month, you should look at your Adwords to make sure everything is going OK.
Traffic sources are either naturally derived by GA, or programmed by you via the utm_ variables.
You should setup Google Analytics goals for all of your KPIs (Key Performance Indicators). These might be something like a registration, or an order, or creation of a review, or anything similar. You'll probably want to look at two things here: a) the number of goal conversions on a month-over-month basis, and b) the funnel conversion % (found on the Goal Funnel page), also month-over-month. Of course, both should be going up, or at least holding steady.
Secondary metrics are still important to keep an eye on, but are less important than monitoring the core.
Bounce rate is the number of people where the entry page and exit page is the same; e.g. they 'bounced' into and out of the site. As a site-wide metric, it isn't terribly useful, although it doesn't hurt to keep an eye on it. As a page-by-page (or even page-by-page sorted by source) basis it is extremely useful. For example, if everybody coming to your product page from your Adwords Ad was bouncing, you might assume that they're not finding what was promised in the ad. See Finding Problem Pages.
One of the things you may want to do is find 'problem' pages on your site. The best way to do this is via the 'Exit %' and 'Bounce Rate' columns on Top Content. Try this: Go to Content → Top Content. Sort by Bounce Rate or Exit % in descending order. Use Advanced Filters to show only pages with more than 100 pageviews (or whatever number makes sense for your site) to get rid of the random noise.
New vs. returning can give you a good snapshot of whether most of your traffic is from new people, or most is from returning visitors. There isn't a right or wrong answer as to what the percentage should be; it will depend on your specific situation.
In my opinion, you should keep an eye on the above … and ignore everything else except for specific things you're trying to optimize on. There are a few things you shouldn't look at:
A pageview is a single human/browser visiting a single page on your site. This makes a great vanity metric, because the number is usually huge, but it doesn't mean a lot. First, GA allows you to create 'virtual pageviews' via JavaScript, so this number is typically inflated. Second ask yourself if a high number of pageviews (or pageviews/visit) is a good thing or a bad thing? If you're a media site, its almost certainly a good thing. But for everybody else? Probably just means your visitors spend a lot of time looking around on your site and not finding what they want.
Time on Site is how long a visitor spent looking at your site. However, it can be misleading for several reasons: a) people often leave browser windows open, increasing the average even when they're not looking at a page. b) Google calculates TOS by subtracting the last timestamp it sees from the first. However, consider the case where somebody looks at just one page - the last timestamp is the same as the first, so the TOS is 00:00:00, even if that user spent time looking at that page! c) Even if you could calculate TOS accurately; what's good? A high TOS could mean people didn't find what they were looking for.
Many of the other metrics that Google provides, such as the geographic breakdown of where people are visiting you from, could be considered 'info porn'. They're cool, and they might be useful in very specific situations, but they're not really useful in terms of keeping track of overall website health.